Fractal boundary basins in spherically symmetricϕ4theory

Abstract

Results are presented from numerical simulations of the flat-space nonlinear Klein-Gordon equation with an asymmetric double-well potential in spherical symmetry. Exit criteria are defined for the simulations that are used to help understand the boundaries of the basins of attraction for Gaussian "bubble" initial data. The first exit criteria, based on the immediate collapse or expansion of bubble radius, is used to observe the departure of the scalar field from a static intermediate attractor solution. The boundary separating these two behaviors in parameter space is smooth and demonstrates a time-scaling law with an exponent that depends on the asymmetry of the potential. The second exit criteria differentiates between the creation of an expanding true-vacuum bubble and dispersion of the field leaving the false vacuum; the boundary separating these basins of attraction is shown to demonstrate fractal behavior. The basins are defined by the number of bounces that the field undergoes before inducing a phase transition. A third, hybrid exit criteria is used to determine the location of the boundary to arbitrary precision and to characterize the threshold behavior. The possible effects this behavior might have on cosmological phase transitions are briefly discussed.

Reissner–Nordstrom–anti–de Sitter nontopological solitons in broken Einstein-Maxwell-Higgs theory

“…Figure 19 shows these solutions colored based on the number of bounces (or modulations), n mod , they undergo before inducing a runaway phase transition. This behavior is similar to the fractal boundary basins observed in real scalar field oscillon dynamics [34]. Figures 20 and 21 show the time evolution of the bubble radius for two different coherent solutions, u 0 = 0.1312 and u 0 = 0.168, for a variety of perturbations, ∆ω.…”

“…Similar to behavior observed in oscillon dynamics [34], PT regions of (∆ω, u 0 ) space with n mod modulations are surrounded by PT regions with (n mod + 1) modulations that approach the region with n mod modulations in a log-periodic fashion. To demonstrate the log-periodic nature of the bounce regions, one first needs to find the boundary of such a region.…”

“…Oscillons created with a double-well potential in flatspace describe "bubble" solutions that are of interest in the study of cosmological phase transitions [32,33] and were shown to have fractal boundaries in the space of possible end states [34]. Self-gravitating scalarons discovered in [35] were shown to be unstable solutions that decayed into either an expanding vacuum bubble or a Schwarzschild black hole.…”

Dynamical Boson Stars

“…An interesting numerical approach to evolving oscillons adopts coordinates that blueshift and damp outgoing radiation of the massive scalar field [115, 117]. A detailed look at the long term dynamics of these solutions suggests the existence of a fractal boundary in parameter space between oscillatons that lead to expansion of a true-vacuum bubble and those that disperse [116]. …”

Radiation and relaxation of oscillons

“…On the theoretical side of understanding the dynamics of oscillons, there have been a number of studies of oscillons of 4 theory in smallamplitude approximation [10][11][12][13]. Furthermore, the attractor basin of oscillons and its fractal nature have been studied both analytically [14] and numerically [15]. Very recently there has been interest in oscillons coupled to gravity, oscillatons [16][17][18].…”